PATH INTEGRALS FOR QUADRATIC LAGRANGIANS IN TWO AND MORE DIMENSIONS

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1 PATH INTEGRAL FOR QUADRATIC LAGRANGIAN IN TWO AND MORE DIMENION G.. DJORDJEVIC and LJ. NEIC Deartment of Physcs, Faculty of cences, P.O. Box 4, 8 Ns,, Ns, erba, gorandj@juns.n.ac.yu ABTRACT A general form of the kernel of the oerator of evoluton for quadratc lagrangans n two and three sace dmensons s obtaned. Ths formula s nvarant n resect to nterchange of real and -adc number felds. Ths result gves chance to exlore more realstc systems because all known results was related to the one-dmensonal systems. Keywords: Feynman s ath ntegral, -adc quantum mechancs. Introducton The nventon of the ath ntegral method [] s one of the major achevements n quantum theory. Orgnally develoed as a sace-tme aroach to non-relatvstc quantum mechancs, Feynman's ath ntegrals became very soon of great mortance n quantum electrodynamcs. Presently, t s a very useful tool and sometmes nevtable ngredent of many modern hyscal theores (for examle, suerstrng theory and quantum cosmology). An advantage n comarson to the oerator aroach les n the fact that the ath ntegral s more ntutve and comrehensve. It gves a global ont of vew on the roblem n queston, whle (dfferental) oerator aroach offers only a local one. Durng the last 5 years alcatons of -adc numbers and adeles have been attracted a sgnfcant nterest, manly n mathematcal hyscs, the Planck scale theory and the very early unverse (for a revew, see []). There s a common belef that the usual cture of sace-tme as a smooth seudo- Remannan manfold has to be substantally changed around Planck length. Besdes an dea of satal noncommutatvty [3], nonarchmedean sace-tme s also a romsng mathematcal concet for a basc hyscal geometry. From quantum gravty ont of vew, the Planck length s the mnmal one whch can be n rncle measured. But ths result s derved usng concets of archmedean geometry and real numbers. It seems qute natural to take nto account also nonarchmedean geometry based on -adc numbers at the Planck scale. In comlex-valued -adc quantum mechancs, a dfferental aroach s ll defned. However, -adc generalzaton of Feynman's ath ntegraton s qute ossble [,4], and for one-dmensonal systems wth quadratc Lagrangans, a general formula for the roagator s rgorously derved [5]. The roblem of connecton and unfcaton of standard and all -adc quantum mechancs s successfully 6

2 solved n the form of adelc quantum mechancs. Durng the last few years, besdes a few of standard and very nterestng one-dmensonal models, whose dynamcs s descrbed by means of an ntegral kernel K of the evoluton oerator U, some multdmensonal systems have been also treated by the same technque [6]. The man am of ths aer s to evaluate analytcally two- and three-dmensonal -adc ath ntegrals and to fnd a general formula for the corresondng roagator wth quadratc actons. We restrct our consderaton to the systems wth Lagrangans at most quadratc n dynamcal varables, whose classcal solutons are analytc functons. We wll see that hgher-dmensonal generalzatons are also ossble, but, as n real case [7], t s techncally very comlex roblem.. -Adc numbers and analyss Let us note that all numercal exermental results belong to the feld of ratonal numbers Q. The comleton of ths feld wth resect to the standard norm (absolute value) leads to the feld of real numbers R Q. Accordng to the Ostrowsk theorem, besdes absolute value and -adc norms ( s a rme number) there are no other nequvalent and nontrval norms on Q. As n the real case, t s ossble to make algebrac closure of these locally comact number felds Q, but ther structure s more comlcate and reacher. Any -adc number x Q, can be resented as an exanson γ x = x ( x + x + x + ), γ Z, () where x are dgts x =,,,, x. -Adc norm of any term x γ + n () s ( γ + ). The - adc norm s nonarchmedean (ultrametrc) one,.e. x + y max( x, y ), and as a consequence, there are a lot of exotc features of -adc saces. For examle, all onts n a dsc γ Bγ ( a) = { x Q : x a } can be regarded as a center of ths dsc. It leads to the total dsconnectedness of -adc saces. Ths way, a choce of the accetable formalsm n quantum theory s constraned. There s no natural orderng on Q, but one can defne a lnear order as follows: x < y f x < y, or f x = y then an ndex m exsts, such that followng s satsfed: x < = y, x = y,, xm = ym, xm ym. Generally seakng, there are two analyses over Q. One of them s connected wth the ma ψ : Q Q (manly used n classcal sector of the -adc dynamcs), and the second one s related to the ma ψ : Q C (n quantum dynamcs). In the case of -adc valued functons, dervatves of φ (x) are defned as n the real case, but wth resect to the -adc norm. When, the -adc valued ntegrals are n the queston, we deal wth an analytc functon and then an defnte ntegral s = n= n nt φ ( φ, φ, t Q () b n φ n n+ n= n+ n+ φ ( dt = ( b a ). (3) a 6

3 In the case of mang Q C, there s not standard dervatve, and some tyes of seudodfferental oerators have been ntroduced [8]. However, t turns out that there s a well defned ntegral wth the Haar measure. In artcular, t wll be emloyed the Gauss ntegral Q / β ( 4α αx + bx) dx = λ ( α ) β ( ), α (4) where ( u) = ex(π{ u} ) s a -adc addtve character. Here, { u} denotes the fractonal art of u. By defnton λ υ ( α ), υ =,,3, s an arthmetc comlex-valued functon Q λ ( x ( sgna), x Q, (5), γ = k,, x λ ( x) = ( ), γ = k +, (mod 4), x ( ), γ = k +, 3(mod 4), (6) x [+ ( ) ], γ = k, λ ( x) = x + x x ( ) [+ ( ) ], γ = k +, (7) wth the followng basc roertes: λ (, λ( a α) = λ ( α), λ ( α) λ ( β λ ( α + β ) λ ( α + β ), λ ( α) =. (8) multaneous treatment of real and -adc numbers can be realzed by concet of adeles. An adele a A s an nfnte sequence a = ( a, a,, a, ), where a R and a Q wth the restrcton that a Z for all but a fnte set M of rmes. The set of all adeles A can be wrtten n the form A = A( M ), A( M R Q M M Z M. (9) A s a toologcal sace and t s a rng wth resect to comonentwse addton and multlcaton. There s a natural generalzaton of analyss on R and Q to analyss on A. 3. Feynman's ath ntegrals n ordnary quantum mechancs In the begnnng quantum mechancs was bult u by analogy wth the formulaton of classcal mechancs n terms of the Hamltonan functon. In 933. Drac announced the dea of formulaton quantum mechancs on the bass of the Lagrange functon. Ths dea was develoed further by Feynman [] who regarded a quantum mechancal moton from one sace-tme ont ( x ', ) to another one ( x ", ) as a sueroston of motons along all ossble aths connectng two onts. The corresondng robablty amltude s (, ) (, ) ħ <, > = e, () where s an acton along corresondng trajectory. Dynamcal evoluton of any quantum-mechancal system, descrbed by a wave functon ψ ( x,, s gven by ψ ( ) = U (, t ') ψ (, ) = K (, ) ψ (, ) d () and K (, ) s a kernel of the untary evoluton oerator U (, ) (quantum-mechancal roagator). In the Feynman formulaton of quantum mechancs kernel K was ostulated to be the ath ntegral ( ) π, ex q,, dtdq, () (, ) h 63

4 where x ' = q( ) and x " = q( ), and h s the Planck constant. One can easly deduce the followng three general roertes Q K (, x, x, t;, ) dx K *(, ) z,, ) d = δ ( z), Q K ( t;, = δ ( ). For the classcal acton (, ) whch s olynomal quadratc n x and t has been shown [9] that n ordnary one-dmensonal quantum mechancs π, ) = ex. (4) h h It can be rewrtten n the followng form K (, λ (, ), (5) h h h where s an addtve character of the feld of real numbers R. o called van Hove-Morette-Paul ermnant aears n the multdmensonal case. It should be noted that ths formula was comuted by subdvson of tme segment nto very small ntervals. The roagator for the quadratc acton n two-dmensonal case, n our notaton, has form K (,, (,, ). (6) h h D-dmensonal generalzaton of the transton amltude for a quadratc classcal acton contans a ermnant: K where we defned and (, ) x = ( xa ), a =,... D. h xa" x / / ' h xa" x ' / = λ b b (3) (, ), (7) h λ = D sgn, (8) h x " ' a xb h xa" xb' 4. Feynman's ath ntegrals n -adc quantum mechancs The nvestgaton of -adc functonal ntegraton s motvated by a successful alcaton of -adc numbers n many arts of theoretcal and mathematcal hyscs, and artcularly n quantum mechancs. One of the greatest achevements n the use of -adc numbers n hyscs s a formulaton of -adc quantum mechancs. The elements of the corresondng Hlbert sace L ( Q ) are some comlex-valued functons of a -adc argument. Quantzaton s erformed by the Weyl rocedure. Instead of the chrodnger equaton, an egenvalue roblem and dynamcal evoluton of a system are usually defned by means of an untary reresentaton of the evoluton oerator U ( on L ( Q ). We emhasze that n the -adc quantum mechancs does not exst dynamcal dfferental equaton of the chrodnger tye. For ths reason, -adc quantum dynamcs s defned over the kernel K of the evoluton oerator 64

5 U ( ) ψ ( ) = K (, ) ψ (, ) d. (9) Because of that we nvestgate general exresson for ths kernel lke n the standard quantum mechancs. For all roertes whch hold for the kernel K n standard quantum mechancs, there are corresondng -adc counterarts. Of course, all ntegraton n (3) are erformng over the Q. - Adc generalzaton of () was suggested n [4] and one can wrte n the form ( ) ( ) K (, [ q] Dq = q,, dt dq(. () (, ) h (, ) h 5. Two -dmensonal -adc satal case In two-dmensonal -adc case we nvestgate general soluton of the ath ntegral Q ( q, q,,, dt K (,, ) DqDq, () (,, ) for the system wth quadratc Lagrangans (wth resect to the q,q and, ). After Taylor exanson of the acton around classcal ath q (, q(, we rove two-dmensonal -adc analogue of the Feynman s theorem K (,, N (, ) ( (,, ) ). () After alyng roertes (3) for the -adc roagator n two-dmensonal case we have [] = ) x x x y x x x y K x y t x y t ' " ' " ' " ' " ( ", ", "; ', ', ' λ x " y ' y ' y ' ( ). (3) Ths -adc result has the same form (7) as n the real case. 6. Three-dmensonal -adc satal case We extend here our revous one and two-dmensonal nvestgatons of -adc ath ntegrals to the three-dmensonal case (., ) K (,,,, q, q, q3,,, 3, dtdqdqdq3, (4) (,,, ) h for a system whch Lagrangan s a quadratc olynomal wth resect to q and =,, 3. Classcal acton s related to the classcal -adc trajectory q (,.e = [ q, q, q3, ] = (,,,, ) q, q, q3,,, 3, dt, (5) wth x " = q ( ), = q ( ), = q3( ), = q( ), = q ( ), = q3( ). Usng the Taylor exanson of [ q, q, q3] around the classcal ath, wth δ [ q, q, q3] =, we obtan three dmensonal -adc analogue of the Feynman s theorem, namely K (,,,, N (, ) ( (,,,, )). (6) Alyng the roerty (3) to the kernel (6), erformng exanson of the classcal acton around the classcal ath, ntegratng over x ', and z ' and usng standard roertes of the δ functon, we obtan t / 65

6 " " x y N (. (7) nce the factor N (, ) can be resented as N (, A(, ) N (, ), the form of the factor A (, ) n three dmensonal case has to be nvestgated. We start wth general form of the quadratc Lagrangan for a system wth three degrees of freedom and try to fnd general soluton of the corresondng classcal equaton of moton. After substtuton of ts solutons n the classcal acton (5) and alyng roertes (3) fnally we have / K (,,,, λ x ' y ' z ' / x ' y ' z ' y ' ( (,,, )) Note ths -adc result n three dmensons has the same form as (7) n the real case. 7. References. (8) [] R. P. Feynman, Rev. Mod. Phys. (948), 367; R. P. Feynman, A. Hbbs, Quantum Mechancs and Path Integrals, McGraw Hll, New York, 965. [] V.. Vladmrov, I. V. Volovch and E. I. Zelenov, -adc Analyss and Mathematcal Physcs, World centfc, ngaore 994. [3] A. Connes, Noncommutatve geometry, Academc Press 994. [4] E. I. Zelenov, J. Math. Phys. 3, (99) 47. [5] G.. Djordjevc and B. Dragovch, On -Adc Functonal Integraton, n Proc. of the II Math. Conf. In Prstna, Prstna (Yugoslava), 996.; G.. Djordjevc, B. Dragovch, Mod. Phys. Lett A, (997) 455; G.. Djordjevc and B. Dragovch, Theor. Math. Phys. Vo. 4, No., () 59.; Djordjevc G.., Dragovch B. and Nesc Lj., Mod. Phys. Lett. A 4, (999) 37.; G. Djordjevc, B. Dragovch and Lj. Nesc, Infnte Dmensonal Analyss, Quantum Probablty and Related Tocs, Vol. 6, No. (3) 79, [6] G.. Djordjevc, B. Dragovch, Lj. Nesc, I.V.Volovch, Int. J. Mod. Phys. A7 () 43 [7] C. C. Groosjean, J. Com. Al. Math. 3 (988) 99 [8] D. D. Dmtrjevc, G.. Djordjevc and Lj. Nesc: Fourer transformaton and seudodfferental oerator wth ratonal art, artcle resented at the Conference BPU5, Vrnjacka Banja 3. [9] C. Morette, Phys. Rev. 8 (95) 848. [] G.. Djordjevc, B. Dragovch and Lj. Nesc, -Adc Feynman s ath ntegrals, n Procc. Of the Internatonal Mathematcal Conference Flomat,